JOURNAL OF RESEARCH of the National Bureau of Standards-A. Physics and Chemistry Vol. 65A, No.4, July- August 1961

Electrical Properties and Kinetics of Electrode Reactions Ralph J. Bradd

(April 12, 1961)

A common basi s for invest igations of the properties of electrode reactions is provided. The basic equations of electrostatics and electrodynamics and the assumption that e lec­trode reactIOns are relaxatIOn processes, are used to develop the equations for the electrical behavior of electrode systems. Thus electrode reaction processes are characterized as two states separated by an energy barrier. The application of static and a lternating fi elds to electrode systems is interpreted in terms of th e kinetic parameters of t he electrode reactions. The equations for impedance are applied to silver and cadium electrode systems reported in the li terature. The agreement of experiment and theoretical expectation is excellent. The equations are also applied to t he interpretation of the impedance of LeClanche cells. The kinetic analysis of a simple unimolecuJar reaction is used to illustrate the kinetic in­te rpretation of experi mental information. This simple analysis may be extended to more co mplex reactions.

1. Introduction

The experimental investigation o[ electrode reac­tion mechanisms has followed four schemes of attack: (a) the determination o[ the steady state electrode potential n,s a function of current, (b) the voltage­step or potentiostatic method where the current den­sity is measured as a function of Lime at constant electrode potentin,l, (c) the current-step or galvano­static method where the electrode potentin,l is meas­ured as a [unction o[ time at constant current density and Cd) the measurement of the impedance of tll~ electrode system as a function o[ the frequency of ftn

applied n,lternating field . Th e analysis of method (a) has been summarized in great detail by Bockris

\ [l].l T4e theoretical basis for analysis o[ methods Cb), (c), and (d) , however, is no t complete n,nd in many cases the fundamental relationship between t hese methods is not recognized.

Method (d), in spite of its wide application, has suffered ill the past from several disadvantages. Contamination problems especially on solid elec­trodes are intensified as n,dsorption n,nd desorption of solution impurities give rise to an impedance in par-

r allel with the electrode impedance. Corrosion reac­tions at metal electrodes give rise to impedances which also contribute to the overall electrode imped­ance. The measurement of the high capacitan ce and low resistance of electrode processes also presents some experimental difficulties.

Over the past sixty years the accepted theory for the prediction of the electrical properties of elec­trode systems has been based primarily on the work of Warburg [2]. Other theories which postulate either that the electrode capacitance is a power fun c­tion of the frequency [3] or that a special circuit ele­ment of constant pha e angle () is present [4] do not conform with experimental evidence [5]. Warburg assumed that diffusion of the reacting species to the electrode surface was the cause of electrode polari­zation. His theory predicted that the phase angle was 45° and that the series resistance and capaci-

1 Figures in the brackets indicate the literature references at the end of this paper.

tance of the electrod e varied as W- 1/ 2 where w= 27rj with j the frequency of the alternating field. The mathematics of the original theory of War burg has been improved by many people. The most signifi­cant improvement in recent years removed the re­striction of a constant phase angle [6] . However, even with this 1"e triction removed there are serious discrepancies between the theory and experiment.

These developments all predict a linear dependence of the resistance and reactance with W-1/ 2. While this relation is obeyed in many instances, frequent references are round to "anomalous" dispersions and absorptions. The cause or the disagreement between experiment and theory has been suggested to be sur­race roughness, adsorption and desorption of solution impurities or the presence of a corrosion proces . The point o[ view in this paper is that the basis of earlier theories is unsatisfactory. A new representa­tion for the consideration of electrode processes which includes the earlier theories will be set forth in this paper.

Since electrode processes may be described as in­volving two equilibrium states separated by an energy barrier [7], it is convenient to treat electrode reactions as relaxation processes. The controlling relaxation process may be either a charge transrer reaction (Ag = Ag+ + e), a chemical reaction simul­taneous with, preceding, or succeed ing, the charge transfer reaction, or a diffusional process of one or more of the participants in the reactions at the electrode-solution interface. Data from the existing literature will be used as a test for the treatment of electrode processes given below. It is the main purpose of this paper to demonstrate the application of relaxation theory and methods to impedance meas­urements of the electrode-solution interface.

2. General Relaxation Theory

2.1. Electrical Theory

Recently there has been increased interest in the electrostatic approach to electrochemical kinetics [8]. In particular, an attempt to relate atomic and elec-

275

tronic polarizations to the calculation of oxidation­reduction reaction rates has been made by Marcus [9] with some success. Relaxation theory has been applied to evaluate the reaction rate constants of ionization reactions in aqueous solutions with good success [10].

As a result of these successes in applying relaxa­tion theory to electrolytic systems it is appropriate to attempt to apply the same theoretical considera­tion to the electrode-solution interface. Both the Debye theory [11] which treats polarization arising from the orientation of permanent dipoles and the Wagner-Maxwell theory [12] which treats polariza­tion arising from the accumulation of charges at the interface in heterogeneous dielectrics, lead to energy absorption as characterized in figure 1 for a hypo­thetical case. Electrode processes also lead to energy adsorption at the electrode-solution interface, often in the power frequency range.

PO WER RADIO ATOMIC ELECTRONIC

FREOUE NCY

FIGURE 1. Energy absorption as a function of frequency for a hypothetical system.

While the derivation of the equations for relaxa­tion processes are well known and are available el ewhere [12], a brief account of the development. is in order.

We will define the field quantity E by

E=-VV (v/m) , (1)

where V is electric potential. The vector polariza­tion, P , is defined by

P = D- EOE (coulombs/m2), (2)

where Eo is the permittivity of a vacuum and D is the displacement. The relationship between E and D is given by

D= eE (coulombs/m2), (3)

where e is the permittivity.

tion is in phase with the alternating field. In the other case there is a measurable phase difference between D and E. A phase difference between D and E can be due to any of three factors: d -c con­ductivity, relaxation effects, and resonance effects.

We will examine now in more detail relaxation effects as they affect the energy absorption at the electrode solution interface. A simple model that may be used for the description of this effect is that of a process in which the energy absorption is characterized by a relaxation time. If a constant electric field is applied at the electrode-solution interface we assume that a polarization will result . from a disturbance of the equilibrium distribution I

of the participants in the electrode reactions at the electrode-solution interface. We will also assume as in other relaxation phenomena that the time rate

of change ~~ of P is proportional to the difference

between the final value, P s, and the actual value P [11,12]:

dP 1 Tt=-:; (P.-P) (coulombs/m2/sec), (4)

where T is a constant with the dimensions of time. Since T is a measure of the time lag, it is called the relaxation time. Integrating eq (4) using the con­dition P t=o= P 00 (i.e., P 00 is the instantaneous con­tribution to the polarization) we obtain

In the opposite condition where a static field is suddenly taken away we have

dP 1 Tt=--:; P (coulombs/m2/sec). (6)

H ere we have P t=o= p. - P 00' Integration gives:

P=(P.-p oo)e-t/T (coulombs/m2). (7)

A description of relaxation effects by equations similar to eqs (5) and (7) was first given by Pellat [13]. The subscripts <Xl and s refer to the values of the quantities at times much less or much greater than T .

For alternating field when there is a phase dif­ference between D and E, it is useful to express D and E as complex numbers,

E=Eoe+Jwt (vim) (8)

D = Doe+J(wt-6) (coulombs/m 2) (9)

where j is -J-l, w=27r X the frequency j of the alternating field, t is the time, and 0 is the phase angle between D and E.

When 0 is independent of E we can write

D = e*E (coulombs/m2). (10)

When the electrode-solution interface is subjected to an alternating field , two possibilities arise which depend on the frequency of the field, the temper­ature, and the nature of th e electrode-solution interface. In the first case there is no measurable phase difference between D and E and the pol ariza- where e* IS the complex permittivity. Here we

276

will be most interested in th e impedance, Z of the 10 r------r-------,,--------,--------,

I electrode-solu t ion interface,

Z = R -jXc (ohms), (11)

in which R i the resistance and Xc is the capacitive reactance.

In relaxation theory it is generally assumed that eq (4) is valid for alternating as well as static fields, and hen ce it om be shown that 2

As a result

and

Z R +R s- R oo ( 1 ) = oo 1+ ' Olms . JWT

(12)

(13)

X .=(R s-R oo) 1;:2T2 (ohms). (14)

By plotting the left hand side of eqs (13) and (14) against log W we obtain the plots shown in figures 2 and 3. The maximum in the X c- log W plot is reached when log WT = O. Thus

_ 1 ( - 1) Wma x - - sec . (15) T

We have

R (max) R s+R oo ( h ) 2 0 ms, (16)

.LYc(max) R ,- R oo ( 1 )

2 01m . (17)

The use of ft, logari thmic scale has the advantage that the CUl'ves in figures 2a and b ar e symmetrical about WT = 1.

Another method of representing the curves is to construct an Argand diagram or complex plane locus in whi ch the imaginary part X c of t he impedance is plotted against the real part R, each point correspond­ing to one frequency (13). From eqs (13) and (14) we obtain

(1 )

I : By plotting R versus X c a semicircle must be obtain ed I with a radius (B ,-R oo )/2, its center on the abscissa at a distance (R s+ B oo ) /2 from the origin. The

I semicircle in figure 3 corresponds to valu es taken , from figure 2. For given valu es of B s and R oo the I R, X c curve is completely defined provid ed the equa-

t ions arc valid . The frequency range in which tbe absorp tion occurs has no influence on the B, X c curve. Thus, this method of representation is inde­pendent of the relaxation time, and reveals the in-

1 herent simplicity of the relaxation process.

~s I I I I I I I I I

o I

- 2

I I I I I

w-Ih Rs+Rao

~f -I 0

LOG wr

10 r---------~

0 -2

FrGU RE 2.

5

- I

RS-R~ -2-

: w'l/r

LOG WT

a. The dependence of R on frequency according to eq (13) assuming R .=9, Roo= l and T= 10-3 sec.

b. The dependence of Xc on frequency according to eq (1 4) , assuming R .=9, R oo= l and T= 10-3

sec.

Rs+R~ -------- -2- ---------]

4 -

2 -

FIGURE 3. Relationship between R and X. according to eq (18), fo r R" Roo and T given in fig ure 2.

An experim ental quantity of interest is tan (J where (J = 90 o - o. From eqs (13) and (14) we find

-t (J_ X c_(R ,- R oo)WT. an - R - j Y + R 2 2

l S co W T (19)

I 'The subscripts'" and s refer to the value of the quantities at frequencies far T aking (0 tan (J /OW ) = 0 we find the n1aximUJTI value above or far below 1/2", respectively. The definition Z=I/(jw.*) was used to f . b relate the complex permittivity to the impedance. 0 tan (J IS 0 tained at the frequency

I 277

<-

1.4

1.2 -

1.0

-Q) .8 c 0 t-

I .6 -

.4

.2 -

-I

I I I I I

_ ~!R; ! Wmox-I /T ~R.: \ :

LOG WT

F IGURE 4 . The de pendence of tan 0 on frequen cy according to eq (19) for R., Roo and r given in jigure 2.

--------- .... - - -......

/- ...... , / "-

/ "' / "'

/ "' / \ I \

I \ I \

I \ I \ I \

o ~ __ ~--~--~--~~~--~--~--~~~--~ o R~

h2C. 2

FWU RE 5. Relationship between R and Xc according to eq (26), dejining the parameter, h for R ., Roo and r given in jigure 3.

The curve for h=O is drawn in with dashed lines.

- t an B,max) Rs-R ", 2..jR ,R ", ·

(20)

(21)

In figure 4 t he dependence of tan B on log W is shown for the values of R s, R ", and T used in figures 2 and 3.

In some cases [14] T may have a distribution of values about a most probable value. The fact that T may have a distribution of values in no way in­validates the equations or graphical r epresentations given previously. Rather it t hrows eq (12) into a more general form:

(22)

where h is identified with the angle defined in figure 5. When h= O eq (22) is reduced to eq (12) . The distribut ion function for t he case wh ere h= O is closely appr02 .. imated by a Boltzman distribution function for T .

A point is chosen on the R , X c curve in figure 5 corresponding to a measurement a t a certain fre­quency WI. The distances u and v from this point to the intersection poin ts R ", and R s of the curve with the abscissa respectively are determined as shown in figure 5. The equat ion

(23)

may be used to evaluate Wmax if u, 'i:, WI, and h ar e known. To determine h a plot of the quan tity log (u /v ) versus log W is cons tructed. The result m ust be a straight line with t he slope (I - h) in ord er tha t eq (22) be applicable.

In t his discussion only one process was assumed to be occurring. ~iVherever more than one relaxation process is occurring simultaneously, the total polari­zation is assulTl.ed to b e the sum of the different polarization cont ributions. As a result eq (5) is written in the form

p ,=p .. + 2: (Ps- P .. Ml-e- 1ITr) (coulombs/m2) (24) I r

and eq (12) is written as

Z R +~(Rs-R "' ) r ( h ) = .. L...Jl + ( · ) 0 ros r JWT r

(25)

with

(26)

and

(27) (

where

R a= (Rs-R ",)r (28)

is the contribution to the resistance by the r-th · relaxation process. II eq (25) does no t apply, a more powerful m eans of combining the impedance ' of th e various processes n1.ust be found .

2.2. Kinetic Theory

In section 2.1 , it was assumed that polarization I is the result of a disturbance of the equilibrium distribution of the participants in the electrode ! reaction. An alternate and sometimes more illu- I minating approach to t h e properties of th e electrode- I solution interface is based on the kin etic behavior of the various processes at the electrode-solution interface. In order to discuss the kinetics of an I

I

278

electrode process, it is necessary first to formulate a reaction mechanism. H the electrode process is a simple unimolecular reaction 3 given by

k,

A~A* (29) k,

the reaction may be depicted in terms of free energy and reaction coordinates as shown in figure 6 [15].

The reaction rate constants are related to the free energy of activation and the field strength by

(30)

and

k - " . [ (D..F:+t::..F)-"A(l -{3) f:,rJ>F] ( - I) 2- U O' exp - RT sec

(31)

where Go and Go' arc consla nLs for a given sysLem, t::..F ~ is the free energy 0 r activation , t::..F is Lhe free energy change Jor the process, "A is the nUll1.ber of charges trans ferred in the process, R is t he gas constant, T is the temperature ill degr ees Kelvin , F is F araday's consta nt, (3 is the symmetry facLar , usually assumed to be }f [17] and t::.. </> is t he difference in the inner poLentilLls 0 1" the soluLion aucl elecLrode. YVe may wriLe t he kinetic equat ions for the reactioll directly.

\ / \ I , /

..... _/

T 11-{3)' !<j>F

REACTION COORDINATE

\ \ \ \ /

\ / ',_...-/

A'

dNJ/dt= k2N2- lcIN I

dN2/dt = - lc2N2+ktN I

(32) F IGURE 6. Details oj the jTee energy of activation barrier for a simple unimolecular process.

(33)

where NI and N2 are the co ncentrations of A and A * which is identical wiLh eq (7) an d respectively. If N = 'L,N" it can be shown Lhat

solution of eqs (32) and (33) Lak es Lhe form

NI = Nol + Olle - o.t (34) and

N 2= No2+012e-o.t (35) where

a= k1+lc2 (sec-I) (36)

and NOl and N02 are the equilibrium valu es of NJ and N z resp ectively. Also we note that 011 = - 0 12 .

Oil is directly proportional to the relaxation time and conductance of the process and the potential drop at the electrode surface. Since N z- Nl is pro­portional to the polarization we no te the exponential approach to equilibrium as required in relaxation theory. When a field is suddenly removed, it follows that

P= (Ps-P oo)e- a ' (coulombs/m2), (37) or

P=(Ps-Poo)e- t / T (coulombs/m 2), (38)

a 'l'he analysis of more complex reactions wi1l not be cons idered here. T'he reader is referred to the available li terature for several treat ments of complex reaction kiuetics [16J.

(39)

Thus T is related directly to the reaction rate con­stants for the process. Th e relaxation time T depe nds on t::..F and t::..F ~ as well as t::.. </>. As a r esult T may be expecLed to be a funcLion of temperature and electrode potential.

The resistance Rs-R oo in eqs (1 2) and (25 ) is the resis tance associated with the given rehtxaLion proc­ess. This resistance will be the r esistance 0 1" Lhe electrode process, Ra= (Rs- R oo ); Ra may be sub­stituted directly into the equation ror exchange current density, Jo,

J o= (RT/"AF )(1/AR a) (amp/m2) (40)

where (RT/"AF] is thermal potential expressed in volts and A is the areft.

Either the Argand diagram or the log plots also may be used to evaluate Lhe resistance or the elec­trode process, Ra , and the relaxation tim e, T. In figures 2b and 3, Ra/2 is the maximum height of Lhe curve. The frequency at the maximum is related to T by eq (15). In figure 2a, Ra is the height of th e step and T is found from the frequency at the mid­point of the step.

279

The equation from older theories (2, 6)

R - X c= R. (41 )

is not a reliable method for determining Ra for an electrode process . From eqs (13) and (14) we see that the difference R -Xc may vary from R", to R. depending on whether w is far removed above or below W m• x. Thus the use of eq (41 ) in the older theories is valid only at frequencies far below Wm• x and if R", is zero.

At equilibrium

dN1/dt = dNz/dt = 0. (42)

The rate of the process at equilibrium is the exchange current. Therefore, we may write

and kdk2=Noz/Nol' (44)

From eqs (30) and (3 1) we can also write

Unfortunately even if the free energy chano'e is known, it is not possible at present to evaluate D.1> for electrochemical processes. In certain circumstances a good approximation for D.<j> is given by

(46)

where Ve - V zpc is t he electrode potential with respect to the electrode potential at zero point of charo'e [1 8]. The zero point of charge is very difficult to ev~luate.

An alternate and possibly more fruitful approach to the ratio kdk2 may be to determine the ratio N 02/ NOl as noted in eq (44). The concentrations of the elements of the process may be found by the analysis of the charge distributions. This type of calculation has been given for mercury electrode interfaces by Grahame [19]. At present, it is not possible to extend the analysis of charge distribution to other electrode interfaces except by inference as the zero point of charge for most metal electrodes cannot be accurately evaluated.

If one of the elements of the process is a metal say A , the concentration of A , NOb may be taken a~ the number of surface atoms. Using eq (43 ) and the ex­change current, kl can be found immediately. Then substituting kl and T into eq (39), k2 can be evaluated. Likewise if N 02, but not Not may be determined, eqs (39) and (43) can be used to evaluate kl and k2.

Once the ratio kdk2 has been evaluated at one potential the rate constants for the process may be determined at any other potential by

k;jk;= (kdk2)exp( - t:..<j>'}..F)/RT (47)

where t:..<j>' is the change in t:..<j> from the old to the new electrode potential and kUk~ is the ratio of rate con­stants at the new electrode potential.

3. Applications

Since equations describing the kinetic and elec­trical properties of the electrode-solution interface are now available, the application of the various equations will be illustrated with data obtained from real electrode systems. The impedance of aD-size LeClanche cell constructed at NBS is shown in figure 7. The cell has a paste liner and a bobbin composi­tion by weight of 8 parts African Mn02 ore, 1.24 parts ammoninm chlorid e and 1 part acetylene black . The impedance of the cell was measured on a substi­tution type Wien bridge described by Vinal [20] . The values of the resistance, capacitance, and re­actance of 4 cells are r eported in table 1 for the fre­quency range 50 cis to 50 kc/s at room temperature. From these data the values of Ra, Wm and R", are given in table 2. Only data for cell 1 is shown in figure 7 for illustration. The introduction of three absorp tion r egions was required to reproduce the im­pedance of LeOlanche cells as a function of frequency. The three regions of energy absorption or dispersion were found in figure 7b by curve fitting using the theoretical behavior of the imaginary part of the im­pedance given by eq (27). These three curves are drawn in lightly in figure 7. It was necessary to in­troduce three relaxation processes in order to fit the experimental curve. The analysis of data by curve fitting is a necessity in the absence of other informa­tion on the location of th e absorption regions.

Experiments using pairs of either zinc or man­ganese oxide clectrodes established that the disper­sion with wm• x= 600 cis was associated with the zinc electrode while the other dispersions were asso­ciated with processes at the manganese oxide elec­trodes. Euler and D ehmelt [21] reached the same conclusions except they did not report the dispersion centered about wrnax= 180 kc/s. A summary of the results from other cells in the group manufactured at NBS is given in table 1. In all cases the sum of the theoretical curves for three absorption r egions fit the experimental behavior of the impedance.

Figure 7 illustrates the usefulness of the various I types of data representation. The Argand diagram (fig. 7a) immediately reveals the complete behavior of the impedance. The plot of log W (fig. 7b) reveals the additivity of the imaingary part of the impedance as assumed in eq (27). Each type of data repre­sentation has its usefulness and applications.

The silver-silver ion and cadmium-cadmium ion electrode systems illustrate the use of eq (13) to represent the behavior of the resistive portion of the electrode impedance [22] . In figure 8 the elec­trode resistance versus log w for both sys tems is shown. The results of the analysis of these graphs is given in table 3. The theoretical curves included for comparative purposes were calculated assuming that the values of R. and Ro> and T given in the caption of the figure . The agreement of theoretical calculation and experiment is excellent.

To illustrate the calculations of k t and k2 for unimolecular reactions the analysis of the silver-silver ion system in figure 8a will be given. Exa mination

280

T ABLE 1. R esistance, capacitance, and reactance of D-size L eClanche cells

Cell f=w/2" 50 100 200

L ________________ R ohm ______ __ ____ 0.623 0.453 0.219 C ,,1" ______________ 15100 6110 3815 Xohm ____________ 0.211 0.261 0.209 2 __________ ___ ____ R ohm ______ _____ _ 0.642 0.439 0.306 C "F _____________ _ 12100 5435 3568 X ohm ____________ 0.263 0.294 0.223 3 _________________ R ohm ____________ 0.942 0.519 0. 378 C"F __________ ____ 9350 3962 2510 X ohm ____________ 0.431 0.402 0.317 4 _________________ R ohm _______ ____ _ 0. 488 0.355 0.260 C"F ______________ 16700 7361 4843 X ohrn ____ _______ _ 0. 191 0.216 0.165

TAB LE 2. Interpretation of data from table 1

Cell ------------

Ohm Ohm serl Ohm serl Oh1" sec-t

L __________________ 0. 159 0.518 596 0. 054 6. 9 k 0.020 210 k 2 _____ ______________

. 151 .602 502 .058 6.3 k . 014 190 k 3 _______ ________ __ __

. 160 .608 583 .060 8.2 k .020 J90 k 4 ___________________

.136 . 420 690 . 050 3.1 k . 010 170 k

of figure 8a shows only one dispersion region. Therefore we need consider only one reaction. It is very unlikely that two reactions in the silver system would have the same relaxation time. vVe will assume that the process at the Ag electrode is:

"I Ag~ Ag++e. (48)

kz

We will assume that the concentration of electrons is very large compared to the other concentrations and that the concentration of electrons r emains constant. Thus, eq (48) may be assumed to be a unimolecular reaction. The concentration of silver atoms will be taken as the number of surface atom.s, approximately 6.0 X 1018 atoms/m2, calculated from the silver metal lattice dimensions [23]. Using eq (43 ) and data from Table 3 with Jo= L1 X 102 amp/m2

or a particle flow of 6.9 X 1020 particles/m2/sec.

6.9 X 1020 = (6 X 1018)kl (particles/m2/sec)

From eq (39) with T = 1/8100

8100 = 115 + k2 (sec-I)

kz= 7985 (sec-I).

From eq (44) the concentration of silver ions, N 2, IS

115/7985 = N 2/(6.0 X 10 18)

N 2= 8.6 4 X 10 16 (ions/m2).

Other investigations of the silver-silver ion system have reported the presence of a low frequency

500 1.0 k 2.0 k 5.0 k 10 k 20 k SOk

0.228 0.203 0.185 0. 176 0.171 0.166 0.164 2740 2262 1 12 1320 946 648 298

o 116 0.070 0.044 0.024 0.017 0.012 0.011

0.218 0.191 0.175 0.167 0.156 0.154 0.153 2632 2153 1697 1241 929.8 652.0 260.2

0.121 0.074 0.047 0.026 0. 017 0.012 0.009

0.250 0.210 0. 1 6 0.171 0.163 0.160 0.161 1829 1495 1211 93 t. 0 926.9 534.4 297.7

0.174 0.106 0.066 0.034 0.022 0.015 0.011

0. 180 0. 167 0.159 0.150 0.145 0.142 0.142 3580 2893 2227 1580 1139 05.0 360.2

0.089 0.055 0. 036 0.020 0.014 0.010 0.009

0.3 ,---,----,---,----r---,------.--,

V>

~ ~

><

0.2 / R/2

0.1 -

I 0.3

O ~~~-L-~~--L-~-L--~--~ 0.1 R~ 0.2 0.4 0.5 0.6 0.7

R. OHMS

0.3 ,.-------,------,-------,-----,

0.2

0.1 -

100 IK 10K lOG w

100 K

FIGU RE 7. a. A rgand diagram of the impedance of a LeClanche cell.

b. The dependence of Xc on frequency for a L eClanche cell.

TABLE 3. S w nmary of data from figure 9

Elec-Electrode System Roo R a l T trode To

area

Ohms Ohms sec m'XlO' amp/in' Ag-Ag+

[10 g AgKO, ] per 100 cm' water __ 130 112 1.2,X lO-' 0. 04 1.1X 102

Cd-CdH

[15 g CdSO, ] per 100 cm' water __ 230 90 6.4 X lO-' .04 1AX lO'

I The experlmental cell was composed of two identical electrodes separated by a solution containing ions of the electrode material. The total resistance asso­ciated with the electrode reaction is the sum of the resistance colltribution of each electrode. As a resnlt tbe resistance used in calculati@n is R./2.

281

L

~ ~ ~

=

~

'"' ~ = =

250

Ag - AgNO j

200 -

150

0

100 100 I K 10 K 100 K

lOG w

350

Cd - CdS04

300

250

200 ~ ________ ~ __________ ~ ________ ~ ______ ~

100 I K 10 K LOG w

100 K

FIGU RE 8. a. The dependence of resistance, R , on frequency for the silver-silver i on (10 g AgNO~ per 100 em" of solution) electrode system.

'The solid line was calculated usingeq (25) and ass uming that R.~ 242, R oo~ 130 and T~ 1.23X lO-'. T he circles are the ex perimental data [23J.

b. The dependence of resistance, R , on frequency for the cadmi1,m ion (15 g CdSO, per 100 cm' of solution) electrode system .

~'he solid line was calculated using eq (25) and assuming that R.~320, R oo ~ 230 an d T ~ 6.4X 10-'. 'rhe circles are the experimental data [23].

dispersion l24] associated with a diffusion process. This is not noticed in Banerji's work, figure 8a, used here for illustration [22]. A similar analysis may be made for the cadmium-cadmium ion system.

4 . Concluding Remarks In this paper a point of view of the kinetics and

electrical properties of electrode systems has been taken that differs from previous theories. The description of electrode reactions as relaxation proc­esses, which includes charge-transfer, and chemical and diffusion processes, is very general in nature. Since diffusion processes may be classified as relaxa­tion processes, the representation given in this paper will include previous theories. Equations for the prediction of the electrical behavior of electrode processes have been given. The representation given in this paper based on relaxation phenomena was

applied to the electrical behavior of electrodes previously reported in the literature, namely, Ag, Ag+ and Cd, Cd+; conformity with theoretical predictions was obtained. The representation was also applied successfully to the electrical behavior of LeClanche cells. Only cm'sory observations on the galvanostatic and potentiostatic methods have been made and the problem of nonlinear combinations of electrode processes has not been considered. Also, resonance effects have not been treated in this paper.

The author thanks .M . G. Broadhurst, J. 1. Lauritzen, Jr ., J. D. Hoffman, and F. B . Silsbee for making a number of helpful suggestions during the course of this ·work.

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